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# ss2ss

state-space to state-space conversion, feedback, injection

### Syntax

`[Sl1,right,left]=ss2ss(Sl,T, [F, [G , [flag]]])`

### Arguments

Sl

linear system (`syslin` list) in state-space form

T

square (non-singular) matrix

Sl1, right, left

linear systems (syslin lists) in state-space form

F

real matrix (state feedback gain)

G

real matrix (output injection gain)

### Description

Returns the linear system `Sl1=[A1,B1,C1,D1]` where `A1=inv(T)*A*T, B1=inv(T)*B, C1=C*T, D1=D`.

Optional parameters `F` and `G` are state feedback and output injection respectively.

For example, `Sl1=ss2ss(Sl,T,F)` returns `Sl1` with:

and `right` is a non singular linear system such that `Sl1=Sl*right`.

`Sl1*inv(right)` is a factorization of `Sl`.

`Sl1=ss2ss(Sl,T,0*F,G)` returns `Sl1` with:

and `left` is a non singular linear system such that `Sl1=left*Sl` (`right=Id` if `F=0`).

When both `F` and `G` are given, `Sl1=left*Sl*right`.

• When `flag` is used and `flag=1` an output injection as follows is used

and then a feedback is performed, `F` must be of size `(m+p,n)`

`right` and `left` have the following property:

`Sl1 = left*sysdiag(sys,eye(p,p))*right`
• When `flag` is used and `flag=2` a feedback (`F` must be of size `(m,n)`) is performed and then the above output injection is applied. `right` and `left` have the following property:

`Sl1 = left*sysdiag(sys*right,eye(p,p)))`

### Examples

```Sl=ssrand(2,2,5); trzeros(Sl)       // zeros are invariant:
Sl1=ss2ss(Sl,rand(5,5),rand(2,5),rand(5,2));
trzeros(Sl1), trzeros(rand(2,2)*Sl1*rand(2,2))
// output injection [ A + GC, (B+GD,-G)]
//                  [   C   , (D   , 0)]
p=1,m=2,n=2; sys=ssrand(p,m,n);

// feedback (m,n)  first and then output injection.

F1=rand(m,n);
G=rand(n,p);
[sys1,right,left]=ss2ss(sys,rand(n,n),F1,G,2);

// Sl1 equiv left*sysdiag(sys*right,eye(p,p)))

res=clean(ss2tf(sys1) - ss2tf(left*sysdiag(sys*right,eye(p,p))))

// output injection then feedback (m+p,n)
F2=rand(p,n); F=[F1;F2];
[sys2,right,left]=ss2ss(sys,rand(n,n),F,G,1);

// Sl1 equiv left*sysdiag(sys,eye(p,p))*right

res=clean(ss2tf(sys2)-ss2tf(left*sysdiag(sys,eye(p,p))*right))

// when F2= 0; sys1 and sys2 are the same
F2=0*rand(p,n);F=[F1;F2];
[sys2,right,left]=ss2ss(sys,rand(n,n),F,G,1);

res=clean(ss2tf(sys2)-ss2tf(sys1))```